Bounded moving average

The intuition behind bounding the moving average is that the more the price falls, the more the moving average falls and the more the price and the moving average are low relative to before. Similarly, the more the price rises, the more the moving average rises and the more the price and the moving average are high relative to before. Mapping the moving average onto [−1, 1] where –1 corresponds to low and 1 corresponds to high constrains the moving average to move within a fixed range. This makes it possible to measure the position of the moving average as defined by its location within [−1, 1] . The bounding algorithm is:

1. Compute the n-day moving average of the price mat n

(pt)

2. Set the minimum of the price and mat n (pt) to bt − _=min(p t,mat n (pt))

3. Set the maximum of the price and mat n (pt) to bt + _=max(p t,mat n (pt)) Since bt − _{≤ p} t ≤bt

+ _{for all t, it holds that ma}

t n (bt −_)≤ma t n (pt)≤mat n (bt +_{) :}

4. Compute the n-day moving average of bt

− _{and set this to the lower bound ma}

t n

(bt

−₎

5. Compute the n-day moving average of bt

+ _{and set this to the upper bound ma}

t n

(bt

+₎

The moving average is then normalised to fit [−1, 1] :

6. Define the bounded moving average as φ(mat n (pt))= 2(ma_tn (p_t)−ma_tn (b_t−)) mat n (bt +_)−ma t n (bt −₎ −1

Figure 4.1 plots an example. The minima and maxima of Figure 4.1 are especially important. Mathematically, when the price crosses its average, the moving average changes direction. If the price cuts up through its average from below, the moving average changes direction from falling to rising. Similarly, if the price cuts down through its average from above, the moving average changes direction from rising to falling. This means that the minima and maxima in the moving average of Figure 4.1 are identical to the buy/sell signals generated by the price crossover rule defined in Chapter 1. Further, the minima and maxima in the moving average are identical to the minima and maxima in the bounded moving average. Hence, the position of the buy/sell signals is known. This information can be used to remodel the price crossover rule by changing the way it responds to the position of the buy/sell signals and is the approach

underlying the trade reduction rule. Given that their buy/sell signals are not generated by the price crossing its average, there is no equivalent for the other types of trading rule defined in Chapter 1. For the other types of trading rule, the bounded moving average could also follow a path similar to Figure 4.1 as the result of a single trade. This means that it is hard to identify patterns that can be defined as general case. However, it is possible to capture something of the nature of the other types of trading rule provided not all minima and maxima result in a buy/sell signal. A feature of the trade reduction rule is that it achieves this without a second moving average.

Figure 4.1 Example of the moving average and its mapping to the bounded moving average.

Figure 4.2 plots an example frequency distribution of the position of the minima and maxima within [−1, 1] . The distribution is non-normal and heavily weighted in the left and right tails. Minima tend to occur close to –1 and maxima tend to occur close to 1. The reason for this is that the bounded moving average is defined self-referentially. The bounded moving average measures the extent to which the price goes up and down in terms of the similarity between

Moving Average 100 200 300 400 500

Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97

Time

Price

price

moving average

Bounded Moving Average

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97

Time

the moving average and itself. As a result, the clustering in the tails is due to persistence in direction. The longer the price continues in the same direction, the more likely it is for the moving average to resemble itself. The more likely it is for the moving average to resemble itself, the more likely it is for the bounded moving average to approach and settle on ±1. The U-shape is due to the bounding algorithm. The same U-shape is found in a random walk with the difference in dynamics reflected in differences in frequency. Random walks are found to be less dense in the left and right tails and more heavily weighted through the middle.

Figure 4.2 Example frequency distribution of the position of minima and maxima.